Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution

Abstract

We find empirically that the value of Feynman integrals follows a $\log$-$\Gamma$ distribution at large loop order. This result opens up a new avenue towards the large-order behavior in perturbative quantum field theory. Our study of the primitive contribution to the scalar ${\varphi }^4$ beta function in four dimensions up to 17 loops provides accompanying evidence. Guided by instanton considerations, we discuss the extrapolation of this contribution to all loop orders.

16 pages, v2: various comments and suggestions addressed; v3: version to appear in JHEP

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